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import Mathlib.Data.Set.Basic
import Mathlib.Data.Set.Lattice
import Bookshelf.Enderton.Set.Chapter_1
import Common.Logic.Basic
import Common.Set.Basic
/-! # Enderton.Chapter_2
Axioms and Operations
-/
namespace Enderton.Set.Chapter_2
/-- ### Exercise 3.1
Assume that `A` is the set of integers divisible by `4`. Similarly assume that
`B` and `C` are the sets of integers divisible by `9` and `10`, respectively.
What is in `A โฉ B โฉ C`?
-/
theorem
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1{1, 2, 3}
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1{2, 3, 4}{3, 4, 5}))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌ{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌ{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌ{ mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))โจยฌยฌ{ mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌ{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4}โจยฌยฌ{ mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4}โจยฌยฌ{ mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4}โจยฌยฌ{ mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: ยฌx=2โงยฌxโfun b => b=3โจb=4
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5 hzโ: x=5 hzโ: xโ{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5 hzโ: x=4 hzโ: xโ{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5 hzโ: x=4 hzโ: 4โ{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5 hzโ: x=5 hzโ: 5=2
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5 hzโ: x=4 hzโ: 4โ{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: (ยฌx=2โงยฌxโfun b => b=3โจb=4) โจx=3โจxโfun b => b=4โจb=5 hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=3โจxโfun b => b=4โจb=5 hzโ: xโfun b => b=4โจb=5
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1{1, 2, 3}
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1{2, 3, 4}{3, 4, 5}))
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
x=1 โ
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
xโ{3} โ
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
x=1 โ
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
x=1 โ
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
xโ{3} โ
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
x=1 โ
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
ยฌ{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))
x=1 โ
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: xโ{1, 3} hy: x=1 hz: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))
x=1 โ
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: xโ{1, 3} hy: x=1 hz: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: xโ{1, 3} hy: x=1 hz: { mem := fun as => sa }.11
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: xโ{1, 3} hy: x=1 hz: { mem := fun as => sa }.11
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: xโ{1, 3} hy: x=1 hz: { mem := fun as => sa }.11
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))
x=1 โ
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: xโ{1, 3} hy: x=1 hz: { mem := fun as => sa }.11
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจ{ mem := fun as => sa }.1bs }.12
({ insert := fun asb => b=aโจ{ mem := fun as => sa }.1bs }.13{4}))
({ insert := fun asb => b=aโจ{ mem := fun as => sa }.1bs }.13
({ insert := fun asb => b=aโจ{ mem := fun as => sa }.1bs }.14{5})))
x=1 โ
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: xโ{1, 3} hy: x=1 hz: { mem := fun as => sa }.11
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจ{ mem := fun as => sa }.1bs }.12
({ insert := fun asb => b=aโจ{ mem := fun as => sa }.1bs }.13 ({ singleton := fun ab => b=a }.14)))
({ insert := fun asb => b=aโจ{ mem := fun as => sa }.1bs }.13
({ insert := fun asb => b=aโจ{ mem := fun as => sa }.1bs }.14 ({ singleton := fun ab => b=a }.15))))
x=1 โ
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
xโ{3} โ
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
xโ{3} โ
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
xโ{3} โ
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
ยฌ{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))
xโ{3} โ
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: xโ{1, 3} hy: xโ{3} hz: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))
xโ{3} โ
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: xโ{1, 3} hy: xโ{3} hz: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) hzr: ยฌ{ mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))
xโ{3} โ
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: xโ{1, 3} hy: xโ{3} hz: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) hzr: ยฌ{ mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: xโ{1, 3} hy: xโ{3} hz: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) hzr: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: xโ{1, 3} hy: xโ{3} hz: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) hzr: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: xโ{1, 3} hy: xโ{3} hz: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) hzr: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5}
xโ{3} โ
{ mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4}))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1{1, 2, 3}{2, 3, 4})
{3, 4, 5})
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5})))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌ{ mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})) hc: ยฌ{ mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌ{ mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})) hc: ยฌ{ mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5}
h.mp.right
xโ{ insert := fun asb => b=aโจbโs }.12{3} โ xโ{1}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5}
h.mp.right
xโ{ insert := fun asb => b=aโจbโs }.12{3} โ xโ{1}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5}
h.mp.right
xโ{ insert := fun asb => b=aโจbโs }.12{3} โ xโ{1}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5}
h.mp.right
xโ{ insert := fun asb => b=aโจbโs }.12{3} โ xโ{1}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5} hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5}
h.mp.right
xโ{ insert := fun asb => b=aโจbโs }.12{3} โ xโ{1}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5} hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5} hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5}
h.mp.right
xโ{ insert := fun asb => b=aโจbโs }.12{3} โ xโ{1}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5} hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5} hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5} hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5} hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: x=2
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5} hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5}
h.mp.right
xโ{ insert := fun asb => b=aโจbโs }.12{3} โ xโ{1}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5} hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5} hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5} hy: xโ{ insert := fun asb => b=aโจbโs }.12{3} hz: xโ{3}
A, B, C: Setโ hA: A={1, 2, 3} hB: B={2, 3, 4} hC: C={3, 4, 5} x: โ hx: { mem := fun as => sa }.1x
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ sdiff := fun sta => { mem := fun as => sa }.1asโงยฌ{ mem := fun as => sa }.1at }.1
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3}))
({ insert := fun asb => b=aโจbโs }.12 ({ insert := fun asb => b=aโจbโs }.13{4})))
({ insert := fun asb => b=aโจbโs }.13 ({ insert := fun asb => b=aโจbโs }.14{5}))) ha: { mem := fun as => sa }.1x
({ insert := fun asb => b=aโจbโs }.11 ({ insert := fun asb => b=aโจbโs }.12{3})) hb: ยฌx=2โงยฌxโ{ insert := fun asb => b=aโจbโs }.13{4} hc: ยฌx=3โงยฌxโ{ insert := fun asb => b=aโจbโs }.14{5} hy: xโ{ insert := fun asb => b=aโจbโs }.12{3}
/-! ### Exercise 4.17
Show that the following four conditions are equivalent.
(a) `A โ B`
(b) `A - B = โ `
(c) `A โช B = B`
(d) `A โฉ B = A`
-/
theorem
exercise_4_17_i: โ {ฮฑ : Type u_1} {A B : Setฮฑ}, AโB โ A\B=โ
/-- ### Exercise 4.25
Is `A โช (โ ๐)` always the same as `โ { A โช X | X โ ๐ }`? If not, then under
what conditions does equality hold?
-/
theorem